Categories of frame-completions and join-specifications

Abstract

Given a poset P , a join-specification U for P is a set of subsets of P whose joins are all defined. The set IU of downsets closed under joins of sets in U forms a complete lattice, and is, in a sense, the free U-join preserving join-completion of P . The main aim of this paper is to address two questions. First, given a join-specification U, when is IU a frame? And second, given a poset P , what is the structure of its set of framegenerating join-specifications?To answer the first question we provide a number of equivalent conditions, and we use these to investigate the second. In particular, we show that the set of frame-generating join-specifications for P forms a complete lattice ordered by inclusion, and we describe its meet and join operations. We do the same for the set of 'maximal' such join-specifications, for a natural definition of 'maximal'. We also define functors from these lattices, considered as categories, into a suitably defined category of framecompletions of P , and construct right adjoints for them.